A171498 -id:A171498 - cp's OEIS frontend

A171499 a(n) = 6a(n-1) - 8a(n-2) for n > 1; a(0) = 3, a(1) = 14.

3, 14, 60, 248, 1008, 4064, 16320, 65408, 261888, 1048064, 4193280, 16775168, 67104768, 268427264, 1073725440, 4294934528, 17179803648, 68719345664, 274877644800, 1099511103488, 4398045462528, 17592183947264, 70368739983360

Offset: 0

Klaus Brockhaus, Dec 10 2009

Binomial transform of A171498; second binomial transform of A171497; third binomial transform of A010703.
Related to sequences A001969 and A000069, sum of each group with exponent 1. - Eric Desbiaux, Jul 24 2013
a(n) in base 2 is n+2 1s followed by n 0s. - Hussam al-Homsi, Oct 12 2021

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (6,-8).

Cf. A010703, A171472, A171473, A171476, A171497, A171498.

[4*4^n-2^n: n in [0..30]]; // Vincenzo Librandi, Jul 18 2011

(* This program shows how A171499 arises from the Vandermonde determinant of (1,2,4,...,2^(n-1)). *)
f[j_]:= 2^j - 1; z = 15;
v[n_]:= Product[Product[f[k] - f[j], {j,k-1}], {k,2,n}]
d[n_]:= Product[(i-1)!, {i,n}]
Table[v[n], {n,z}]                     (* A203303 *)
Table[v[n+1]/v[n], {n,z}]              (* A002884 *)
Table[v[n]*v[n+2]/(2*v[n+1])^2, {n,z}]  (* A171499 *)
(* Clark Kimberling, Jan 02 2011 *)
LinearRecurrence[{6,-8},{3,14},30] (* Harvey P. Dale, Sep 05 2021 *)

{m=23; v=concat([3, 14], vector(m-2)); for(n=3, m, v[n]=6*v[n-1]-8*v[n-2]); v}

[4^(n+1) -2^n for n in range(31)] # G. C. Greubel, Aug 31 2023

a(n) = 4*4^n - 2^n = 2^n * (2^(n+2) - 1).
G.f.: (3-4*x)/((1-2*x)*(1-4*x)).
a(n) = 4*a(n-1) + 2^n for n > 0. - Vincenzo Librandi, Jul 18 2011
a(n) = A171476(n+1)/2. - Hussam al-Homsi, Jun 06 2021
E.g.f.: 4*exp(4*x) - exp(2*x). - G. C. Greubel, Aug 31 2023

A237930 a(n) = 3^(n+1) + (3^n-1)/2.

Original entry on oeis.org

3, 10, 31, 94, 283, 850, 2551, 7654, 22963, 68890, 206671, 620014, 1860043, 5580130, 16740391, 50221174, 150663523, 451990570, 1355971711, 4067915134, 12203745403, 36611236210, 109833708631, 329501125894, 988503377683, 2965510133050, 8896530399151

Offset: 0

Philippe Deléham, Feb 16 2014

a(n-1) agrees with the graph radius of the n-Sierpinski carpet graph for n = 2 to at least n = 5. See A100774 for the graph diameter of the n-Sierpinski carpet graph.
The inverse binomial transform gives 3, 7, 14, 28, 56, ... i.e., A005009 with a leading 3. - R. J. Mathar, Jan 08 2020
First differences of A108765. The digital root of a(n) for n > 1 is always 4. a(n) is never divisible by 7 or by 12. a(n) == 10 (mod 84) for odd n. a(n) == 31 (mod 84) for even n > 0. Conjecture: This sequence contains no prime factors p == {11, 13, 23, 61 71, 73} (mod 84). - Klaus Purath, Apr 13 2020
This is a subsequence of A017209 for n > 1. See formula. - Klaus Purath, Jul 03 2020

			Ternary....................Decimal
10...............................3
101.............................10
1011............................31
10111...........................94
101111.........................283
1011111........................850
10111111......................2551
101111111.....................7654, etc.

Colin Barker, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Graph Radius.
Eric Weisstein's World of Mathematics, Sierpinski Carpet Graph.
Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A000244, A003462, A005009, A005032 (first differences), A017209, A060816, A100774, A108765 (partial sums), A199109, A329774.

Cf. A024023, A029858, A057198, A116952, A171498.

[3^(n+1) + (3^n-1)/2: n in [0..40]]; // Vincenzo Librandi, Jan 09 2020

(* Start from Eric W. Weisstein, Mar 13 2018 *)
Table[(7 3^n - 1)/2, {n, 0, 20}]
(7 3^Range[0, 20] - 1)/2
LinearRecurrence[{4, -3}, {10, 31}, {0, 20}]
CoefficientList[Series[(3 - 2 x)/((x - 1) (3 x - 1)), {x, 0, 20}], x]
(* End *)

Vec((3 - 2*x) / ((1 - x)*(1 - 3*x)) + O(x^30)) \\ Colin Barker, Nov 27 2019

G.f.: (3-2*x)/((1-x)*(1-3*x)).
a(n) = A000244(n+1) + A003462(n).
a(n) = 3*a(n-1) + 1 for n > 0, a(0)=3. (Note that if a(0) were 1 in this recurrence we would get A003462, if it were 2 we would get A060816. - N. J. A. Sloane, Dec 06 2019)
a(n) = 4*a(n-1) - 3*a(n-2) for n > 1, a(0)=3, a(1)=10.
a(n) = 2*a(n-1) + 3*a(n-2) + 2 for n > 1.
a(n) = A199109(n) - 1.
a(n) = (7*3^n - 1)/2. - Eric W. Weisstein, Mar 13 2018
From Klaus Purath, Apr 13 2020: (Start)
a(n) = A057198(n+1) + A024023(n).
a(n) = A029858(n+2) - A024023(n).
a(n) = A052919(n+1) + A029858(n+1).
a(n) = (A000244(n+1) + A171498(n))/2.
a(n) = 7*A003462(n) + 3.
a(n) = A116952(n) + 2. (End)
a(n) = A017209(7*(3^(n-2)-1)/2 + 3), n > 1. - Klaus Purath, Jul 03 2020
E.g.f.: exp(x)*(7*exp(2*x) - 1)/2. - Stefano Spezia, Aug 28 2023

A171497 a(n) = 2*a(n-1) for n > 1; a(0) = 3, a(1) = 8.

Original entry on oeis.org

3, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296, 8589934592, 17179869184

Offset: 0

Klaus Brockhaus, Dec 10 2009

8*A000079 preceded by 3.
Binomial transform of A010703.
Inverse binomial transform of A171498.

Index entries for linear recurrences with constant coefficients, signature (2).

Cf. A000079 (powers of 2), A010703 (repeat 3, 5), A171498.

Join[{3},NestList[2#&,8,30]] (* Harvey P. Dale, Dec 17 2011 *)

{m=30; v=concat([3, 8], vector(m-2)); for(n=3, m, v[n]=2*v[n-1]); v}

a(n) = 4*2^n for n > 0; a(0) = 3.
G.f.: (3+2*x)/(1-2*x).
E.g.f.: 4*exp(2*x) - 1. - Stefano Spezia, Aug 04 2024

a(30)-a(32) from Stefano Spezia, Aug 04 2024

A198643 a(n) = 5*3^n-1.

Original entry on oeis.org

4, 14, 44, 134, 404, 1214, 3644, 10934, 32804, 98414, 295244, 885734, 2657204, 7971614, 23914844, 71744534, 215233604, 645700814, 1937102444, 5811307334, 17433922004, 52301766014, 156905298044, 470715894134, 1412147682404, 4236443047214

Offset: 0

Vincenzo Librandi, Oct 28 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Amya Luo, Pattern Avoidance in Nonnesting Permutations, Undergraduate Thesis, Dartmouth College (2024). See p. 11.
Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A027107, A048473, A171498.

Magma
```
[5*3^n-1: n in [0..30]];
```

5*3^Range[0, 30] - 1 (* or *)
NestList[3*# + 2 &, 4, 30] (* Paolo Xausa, Aug 28 2024 *)

a(n)=5*3^n-1 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 3*a(n-1)+2 = 2*A060816(n+1).

G.f.: ( 4-2*x ) / ( (3*x-1)*(x-1) ). - R. J. Mathar, Nov 17 2011

A036543 a(n) = T(3,n), array T given by A048471.

Original entry on oeis.org

1, 9, 33, 105, 321, 969, 2913, 8745, 26241, 78729, 236193, 708585, 2125761, 6377289, 19131873, 57395625, 172186881, 516560649, 1549681953, 4649045865, 13947137601, 41841412809, 125524238433, 376572715305, 1129718145921

Offset: 0

Clark Kimberling

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-3).

n-th difference of a(n), a(n-1), ..., a(0) is 2^(n+2) for n=1, 2, 3, ...

Cf. A146541 (inv. bin. transf.)

[4*3^n-3: n in [0..30]]; // Vincenzo Librandi, Nov 11 2011

4*3^Range[0,25]-3 (* or *) LinearRecurrence[{4,-3},{1,9},25] (* Harvey P. Dale, Aug 16 2011 *)

vector(30, n, n--; 4*3^n-3) \\ G. C. Greubel, Nov 23 2018

[4*3^n-3 for n in range(30)] # G. C. Greubel, Nov 23 2018

Binomial transform of A084242. Second binomial transform of periodic sequence A010688. - Paul Barry, May 23 2003
From Paul Barry, May 23 2003: (Start)
a(n) = 4*3^n - 3;
G.f.: (1+5*x)/((1-x)*(1-3*x));
E.g.f.: 4*exp(3*x) - 3*exp(x). (End)
a(n) = 4*a(n-1) - 3*a(n-2); a(0)=1, a(1)=9. - Harvey P. Dale, Aug 16 2011
a(n) = 3*a(n-1) + 6. - Vincenzo Librandi, Nov 11 2011
a(n) = A171498(n) - 2. - Philippe Deléham, Apr 13 2013

A094615 Triangular array T of numbers generated by these rules: 1 is in T; and if x is in T, then 2x+1 and 3x+2 are in T.

Original entry on oeis.org

1, 3, 5, 7, 11, 17, 15, 23, 35, 53, 31, 47, 71, 107, 161, 63, 95, 143, 215, 323, 485, 127, 191, 287, 431, 647, 971, 1457, 255, 383, 575, 863, 1295, 1943, 2915, 4373, 511, 767, 1151, 1727, 2591, 3887, 5831, 8747, 13121, 1023, 1535, 2303, 3455, 5183, 7775, 11663, 17495, 26243, 39365

Offset: 0

Clark Kimberling, May 14 2004

To obtain row n from row n-1, apply 2x+1 to each x in row n-1 and then put -1+2*3^n at the end. Or, instead, apply 3x+2 to each x in row n-1 and then put -1+2^(n+1) at the beginning.

Subtriangle of the triangle in A230445. - Philippe Deléham, Oct 31 2013

			Triangle begins:
  n\k|   1    2    3    4    5    6     7
  ---+-----------------------------------
  0  |   1;
  1  |   3,   5;
  2  |   7,  11,  17;
  3  |  15,  23,  35,  53;
  4  |  31,  47,  71, 107, 161;
  5  |  63,  95, 143, 215, 323, 485;
  6  | 127, 191, 287, 431, 647, 971, 1457;

Michel Marcus, Rows n=0..99 of triangle, flattened

Cf. A094616 (row sums), A094617, A230445.

Cf. A048473, A171498, A198644

tabl(nn) = {my(row = [1], nrow); for (n=1, nn, print (row); nrow = vector(n+1, k, if (k<=n, (2*row[k]+1), -1+2*3^n)); row = nrow;);} \\ Michel Marcus, Nov 14 2020

T(n,0) = -1+2^(n+1) = A000225(n+1).
T(n,n) = -1+2*3^n = A048473(n).
T(2n,n) = -1+2*6^n.
T(n,k) = -1 + 2^(n+1-k)*3^k. - Lamine Ngom, Feb 10 2021

Offset 0 and more terms from Michel Marcus, Nov 14 2020

A238976 a(n) = ((3^(n-1)-1)^2)/4.

Original entry on oeis.org

0, 1, 16, 169, 1600, 14641, 132496, 1194649, 10758400, 96845281, 871666576, 7845176329, 70607118400, 635465659921, 5719195722256, 51472775849209, 463255025689600, 4169295360346561, 37523658630539536, 337712928837117289, 3039416363020840000, 27354747277647913201, 246192725530212278416

Offset: 1

Kival Ngaokrajang, Mar 07 2014

If the Cantor square fractal is modified as shown in the illustration (see Links), then 4*a(n) is the total number of holes in the modified Cantor square fractal after n iterations. The total number of sides (outside) is 4*A171498(n-1). The total length of the sides (outside) converges to 20 when the initial total side length is 12 (starting with 5 unit squares).
For the Cantor square fractal, the total number of sides (outside) is 4*A168616(n+2). The total number of holes is 4*A060867(n-1) for n > 1. The total length of the sides (outside) converges to 12 with the same initial condition (i.e., 5 unit square); its maximum is 17.333... and is reached at n = 2, 3. The Cantor square fractal and modified one are not true fractals.
See illustrations in links.

Kival Ngaokrajang, Illustrations of initial terms
Eric Weisstein's World of Mathematics, Cantor Square Fractal
Index entries for linear recurrences with constant coefficients, signature (13,-39,27)

Cf. A024023, A060867, A060868, A168616, A171498.

a(n) = ((3^(n-1)-1)^2)/4; \\ Joerg Arndt, Mar 08 2014

a(n) = (A024023(n-1))^2/4.

G.f.: x*(3*x + 1)/((1-x)*(1-3*x)*(1-9*x)). - Ralf Stephan, Mar 14 2014

A321003 a(n) = 2^n(43^n-1).

Original entry on oeis.org

3, 22, 140, 856, 5168, 31072, 186560, 1119616, 6718208, 40310272, 241863680, 1451186176, 8707125248, 52242767872, 313456640000, 1880739905536, 11284439564288, 67706637647872, 406239826411520, 2437438959517696, 14624633759203328, 87747802559414272

Offset: 0

N. J. A. Sloane, Nov 01 2018

Conjectured to be the sum of A175046(i) for 1 <= i < 2^(n+1).

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-12).

Cf. A175046, A171498, A321002.

a := n -> 2^n*(4*3^n-1):
seq(a(n), n = 0 .. 25); # Stefano Spezia, Nov 02 2018

a[n_]:=2^n*(4*3^n-1); Array[a, 25, 0] (* or *)
CoefficientList[Series[-E^(2 x) + 4 E^(6 x), {x, 0, 25}], x]*Table[k!, {k, 0, 25}] (* Stefano Spezia, Nov 02 2018 *)

Vec((3 - 2*x) / ((1 - 2*x)*(1 - 6*x)) + O(x^25)) \\ Colin Barker, Nov 02 2018

a(n) = 2^n*(4*3^n-1); \\ Michel Marcus, Nov 02 2018

From Colin Barker, Nov 02 2018: (Start)
G.f.: (3 - 2*x) / ((1 - 2*x)*(1 - 6*x)).
a(n) = 8*a(n-1) - 12*a(n-2) for n>1.
(End)
E.g.f.: -exp(2*x)+4*exp(6*x). - Stefano Spezia, Nov 02 2018

A171495 a(n) = 3*a(n-1)+4 for n > 0; a(0) = 6.

Original entry on oeis.org

6, 22, 70, 214, 646, 1942, 5830, 17494, 52486, 157462, 472390, 1417174, 4251526, 12754582, 38263750, 114791254, 344373766, 1033121302, 3099363910, 9298091734, 27894275206, 83682825622, 251048476870, 753145430614, 2259436291846

Offset: 0

Klaus Brockhaus, Dec 10 2009

nonn

Binomial transform of A171494; second binomial transform of A010726.

Inverse binomial transform of A171496.

Index entries for linear recurrences with constant coefficients, signature (4, -3).

Equals 2*A171498.

Cf. A010726 (repeat 6, 10), A171494, A171496.

NestList[3#+4&,6,30] (* Harvey P. Dale, Aug 25 2019 *)

{m=25; v=concat([6], vector(m-1)); for(n=2, m, v[n]=3*v[n-1]+4); v}

a(n) = 2*(4*3^n-1).

G.f.: 2*(3-x)/((1-x)*(1-3*x)).

A198646 a(n) = 11*3^n-1.

Original entry on oeis.org

10, 32, 98, 296, 890, 2672, 8018, 24056, 72170, 216512, 649538, 1948616, 5845850, 17537552, 52612658, 157837976, 473513930, 1420541792, 4261625378, 12784876136, 38354628410, 115063885232, 345191655698, 1035574967096, 3106724901290

Offset: 0

Vincenzo Librandi, Oct 28 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A027107, A048473, A171498.

Magma
```
[11*3^n-1: n in [0..30]]
```

11*3^Range[0, 30] - 1 (* Wesley Ivan Hurt, Oct 02 2021 *)

a(n) = 3*a(n-1)+2, a(0)=10.
G.f.: (10-8*x) / ((3*x-1)*(x-1)). - R. J. Mathar, Oct 30 2011
a(n) = 4*a(n-1)-3*a(n-2). - Wesley Ivan Hurt, Oct 02 2021
a(n) = 2*A052909(n+1). - R. J. Mathar, Apr 07 2022

cp's OEIS Frontend

A171499 a(n) = 6*a(n-1) - 8*a(n-2) for n > 1; a(0) = 3, a(1) = 14.

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A237930 a(n) = 3^(n+1) + (3^n-1)/2.

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A171497 a(n) = 2*a(n-1) for n > 1; a(0) = 3, a(1) = 8.

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A198643 a(n) = 5*3^n-1.

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Magma

Mathematica

PARI

Formula

A036543 a(n) = T(3,n), array T given by A048471.

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Magma

Mathematica

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A094615 Triangular array T of numbers generated by these rules: 1 is in T; and if x is in T, then 2x+1 and 3x+2 are in T.

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PARI

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A238976 a(n) = ((3^(n-1)-1)^2)/4.

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A171499 a(n) = 6a(n-1) - 8a(n-2) for n > 1; a(0) = 3, a(1) = 14.

A321003 a(n) = 2^n(43^n-1).